We investigate the Hamilton-Jacobi equation of a probe particle moving on
d-dimensional generalized Lense-Thirring metric. This space-time is different
from the slowly rotating Myers-Perry black hole at second order in rotation
parameters. We show that the dynamics of the probe particle along the time-like
geodesic of the generalized Lense-Thirring space-time is super-integrable and
has more constants of motion with respect to the same dynamics on Myers-Perry
black hole. We also discuss the second rank Killing tensors associated with the
generalized Lense-Thirring metric.